Let G be a linear algebraic group. Informally, a versal G-torsor is a G-torsor (over a field) from which one can recover any other G-torsor. Versal torsors behave similarly to universal objects, but fail to satisfy certain uniqueness properties. In this talk, I introduce several variations of the notion of versality for a G-variety X. These notions have equivalent formulations as properties of twisted forms of X: existence of k-points, density of k-points, k-unirationality and so on. This is joint work with Zinovy Reichstein.